Lemma 2.2 ([1]}) 

\(\mu \)  satisfies \(\mathcal {T}_{1}^d (C)\)  if and only if for all Lipschitz function \(f\)  with \(<f>_{\mu }:= \mathbb {E}_{\mu } f\) , it holds that
\( & \int e^{\lambda (f- <f>_{\mu })} d\mu \le \exp (\frac{\lambda ^2}{2}C \Vert f\Vert _{L(d)} ^2), \hspace{15.0pt} \text{where~~~~} \Vert f\Vert _{L(d)}:= \sup _{x \ne y} \frac{|f(x)-f(y)|}{d(x,y)}.\) 

